Bill Gosper: Then I bashed on it so hard that it got unnervingly ungrotesque: (*) EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] How could anything this simple be new? WDS: Let s=Sqrt[1-t]. Then (*) is rewritten as EllipticK[ 1/2 - (1+s^2)/(4s) ] == EllipticK[ -(1-s)^2/(4s) ] == s^(1/2) EllipticK[1-s^2] Now let s=1-m. Then this is EllipticK[ -m^2/(4(1-m)) ] == (1-m)^(1/2) EllipticK[(2-m)m]. These look kind of like the known identities here http://functions.wolfram.com/EllipticIntegrals/EllipticK/17/01/ but not the same. Perhaps yours arises by composing one of them, with a quadratic transformation. The final form of it I wrote down makes it clear how the argument (if m is small) gets to be quadratically way small. Thus, this offers a very fast way to compute EllipticK. Also, to compute pi/2=EllipticK[0] if we happened to know some other EllipticK special value. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)